#6675220

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1781 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(5,6)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22,26,31)(23,27,35,44)(24,29,36,47)(25,38,40,30)(28,39,45,37)(32,46,33,42)(34,50,51,43)(41,49,52,48);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(21,23)(22,27)(24,32)(25,34)(26,35)(28,41)(29,46)(30,43)(31,44)(33,36)(37,48)(38,50)(39,49)(40,51)(42,47)(45,52);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,24)(22,28)(23,32)(25,39)(26,36)(27,41)(29,38)(30,47)(31,45)(33,35)(34,49)(37,40)(42,43)(44,52)(46,50)(48,51);(6,13)(11,15)(12,14)(19,20)(21,24)(22,29)(23,33)(25,37)(26,36)(27,42)(28,38)(30,45)(31,47)(32,35)(34,49)(39,40)(41,43)(44,46)(48,51)(50,52);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(24,36)(28,45)(29,47)(32,33)(37,39)(41,52)(42,46)(48,49)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582416);